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2*x^2+y^2-8*x+6*y+1=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 2*x + y - 8*x + 6*y + 1 = 0
$$\left(6 y + \left(- 8 x + \left(2 x^{2} + y^{2}\right)\right)\right) + 1 = 0$$
Solución detallada
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 2$$
$$b = -8$$
$$c = y^{2} + 6 y + 1$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{- 8 y^{2} - 48 y + 56}}{4} + 2$$
$$x_{2} = 2 - \frac{\sqrt{- 8 y^{2} - 48 y + 56}}{4}$$
a*x^2 + b*x + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 2$$
$$b = -8$$
$$c = y^{2} + 6 y + 1$$
, entonces
D = b^2 - 4 * a * c =
(-8)^2 - 4 * (2) * (1 + y^2 + 6*y) = 56 - 48*y - 8*y^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{- 8 y^{2} - 48 y + 56}}{4} + 2$$
$$x_{2} = 2 - \frac{\sqrt{- 8 y^{2} - 48 y + 56}}{4}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(6 y + \left(- 8 x + \left(2 x^{2} + y^{2}\right)\right)\right) + 1 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 4 x + \frac{y^{2}}{2} + 3 y + \frac{1}{2} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = \frac{y^{2}}{2} + 3 y + \frac{1}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = \frac{y^{2}}{2} + 3 y + \frac{1}{2}$$
$$\left(6 y + \left(- 8 x + \left(2 x^{2} + y^{2}\right)\right)\right) + 1 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 4 x + \frac{y^{2}}{2} + 3 y + \frac{1}{2} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = \frac{y^{2}}{2} + 3 y + \frac{1}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = \frac{y^{2}}{2} + 3 y + \frac{1}{2}$$
Gráfica
Respuesta rápida
[src]
_______________________________________________________________________ _______________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 14 - 12*re(y) - 2*re (y) + 2*im (y)/| 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 14 - 12*re(y) - 2*re (y) + 2*im (y)/|
\/ (-12*im(y) - 4*im(y)*re(y)) + \14 - 12*re(y) - 2*re (y) + 2*im (y)/ *cos|---------------------------------------------------------------------| I*\/ (-12*im(y) - 4*im(y)*re(y)) + \14 - 12*re(y) - 2*re (y) + 2*im (y)/ *sin|---------------------------------------------------------------------|
\ 2 / \ 2 /
x1 = 2 - ------------------------------------------------------------------------------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------------------------------------------------------------
2 2
$$x_{1} = - \frac{i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14 \right)}}{2} \right)}}{2} + 2$$
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 14 - 12*re(y) - 2*re (y) + 2*im (y)/| 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 14 - 12*re(y) - 2*re (y) + 2*im (y)/|
\/ (-12*im(y) - 4*im(y)*re(y)) + \14 - 12*re(y) - 2*re (y) + 2*im (y)/ *cos|---------------------------------------------------------------------| I*\/ (-12*im(y) - 4*im(y)*re(y)) + \14 - 12*re(y) - 2*re (y) + 2*im (y)/ *sin|---------------------------------------------------------------------|
\ 2 / \ 2 /
x2 = 2 + ------------------------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------------------
2 2
$$x_{2} = \frac{i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14 \right)}}{2} \right)}}{2} + 2$$
x2 = i*((-4*re(y)*im(y) - 12*im(y))^2 + (-2*re(y)^2 - 12*re(y) + 2*im(y)^2 + 14)^2)^(1/4)*sin(atan2(-4*re(y)*im(y) - 12*im(y, -2*re(y)^2 - 12*re(y) + 2*im(y)^2 + 14)/2)/2 + ((-4*re(y)*im(y) - 12*im(y))^2 + (-2*re(y)^2 - 12*re(y) + 2*im(y)^2 + 14)^2)^(1/4)*cos(atan2(-4*re(y)*im(y) - 12*im(y), -2*re(y)^2 - 12*re(y) + 2*im(y)^2 + 14)/2)/2 + 2)
Suma y producto de raíces
[src]
suma
_______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 14 - 12*re(y) - 2*re (y) + 2*im (y)/| 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 14 - 12*re(y) - 2*re (y) + 2*im (y)/| 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 14 - 12*re(y) - 2*re (y) + 2*im (y)/| 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 14 - 12*re(y) - 2*re (y) + 2*im (y)/|
\/ (-12*im(y) - 4*im(y)*re(y)) + \14 - 12*re(y) - 2*re (y) + 2*im (y)/ *cos|---------------------------------------------------------------------| I*\/ (-12*im(y) - 4*im(y)*re(y)) + \14 - 12*re(y) - 2*re (y) + 2*im (y)/ *sin|---------------------------------------------------------------------| \/ (-12*im(y) - 4*im(y)*re(y)) + \14 - 12*re(y) - 2*re (y) + 2*im (y)/ *cos|---------------------------------------------------------------------| I*\/ (-12*im(y) - 4*im(y)*re(y)) + \14 - 12*re(y) - 2*re (y) + 2*im (y)/ *sin|---------------------------------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
2 - ------------------------------------------------------------------------------------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------------------------------------------------------------- + 2 + ------------------------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------------------
2 2 2 2
$$\left(- \frac{i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14 \right)}}{2} \right)}}{2} + 2\right) + \left(\frac{i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14 \right)}}{2} \right)}}{2} + 2\right)$$
=
4
$$4$$
producto
/ _______________________________________________________________________ _______________________________________________________________________ \ / _______________________________________________________________________ _______________________________________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 14 - 12*re(y) - 2*re (y) + 2*im (y)/| 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 14 - 12*re(y) - 2*re (y) + 2*im (y)/|| | 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 14 - 12*re(y) - 2*re (y) + 2*im (y)/| 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 14 - 12*re(y) - 2*re (y) + 2*im (y)/|| | \/ (-12*im(y) - 4*im(y)*re(y)) + \14 - 12*re(y) - 2*re (y) + 2*im (y)/ *cos|---------------------------------------------------------------------| I*\/ (-12*im(y) - 4*im(y)*re(y)) + \14 - 12*re(y) - 2*re (y) + 2*im (y)/ *sin|---------------------------------------------------------------------|| | \/ (-12*im(y) - 4*im(y)*re(y)) + \14 - 12*re(y) - 2*re (y) + 2*im (y)/ *cos|---------------------------------------------------------------------| I*\/ (-12*im(y) - 4*im(y)*re(y)) + \14 - 12*re(y) - 2*re (y) + 2*im (y)/ *sin|---------------------------------------------------------------------|| | \ 2 / \ 2 /| | \ 2 / \ 2 /| |2 - ------------------------------------------------------------------------------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------------------------------------------------------------|*|2 + ------------------------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------------------| \ 2 2 / \ 2 2 /
$$\left(- \frac{i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14 \right)}}{2} \right)}}{2} + 2\right) \left(\frac{i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 14 \right)}}{2} \right)}}{2} + 2\right)$$
=
2 2 1 re (y) im (y) - + ------ + 3*re(y) - ------ + 3*I*im(y) + I*im(y)*re(y) 2 2 2
$$\frac{\left(\operatorname{re}{\left(y\right)}\right)^{2}}{2} + i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{re}{\left(y\right)} - \frac{\left(\operatorname{im}{\left(y\right)}\right)^{2}}{2} + 3 i \operatorname{im}{\left(y\right)} + \frac{1}{2}$$
1/2 + re(y)^2/2 + 3*re(y) - im(y)^2/2 + 3*i*im(y) + i*im(y)*re(y)